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Topic: Hamel basis

	Orthonormal basis - Wikipedia, the free encyclopedia
		In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is, of magnitude 1.
		An orthogonal basis satisfies the same conditions, without the condition of length 1; it is easy to change the vectors in an orthogonal basis by scalar multiples to get an orthonormal basis, and indeed this is a typical way that an orthonormal basis is constructed, via an orthogonal basis.
		An orthonormal basis is not generally a "basis", i.e., it is not generally possible to write every member of the space as a linear combination of finitely many members of an orthonormal basis.
	www.wikipedia.org /wiki/Orthonormal_basis (576 words)

	Basis (linear algebra) - Wikipedia, the free encyclopedia
		The phrase Hamel basis is sometimes used to refer to a basis as defined above, in which the fact that all linear combinations are finite is crucial.
		What is called an orthonormal basis is a set of mutually orthogonal unit vectors that "span" the space via sometimes-infinite linear combinations.
		An orthonormal basis of an infinite-dimensional Hilbert space is therefore not a Hamel basis.
	en.wikipedia.org /wiki/Basis_(linear_algebra) (1033 words)

	Station Information - Hamel basis
		In mathematics, a Hamel basis of a vector space is a set B of vectors in the space such that
		Every Hamel basis of this space is much bigger than this merely countably infinite set of functions.
		Hamel bases of spaces of this kind are of little if any interest; orthonormal bases of these spaces are important to Fourier analysis.
	www.stationinformation.com /encyclopedia/h/ha/hamel_basis.html (276 words)

	PlanetMath: basis (Site not responding. Last check: 2007-11-07)
		A (Hamel) basis of a vector space is a linearly independent spanning set.
		The fact that every vector space has a Hamel basis is an important consequence of the axiom of choice (in fact, that proposition is equivalent to the axiom of choice.)
		This is version 17 of basis, born on 2001-11-27, modified 2004-04-30.
	planetmath.org /encyclopedia/Basis.html (221 words)

	Orthonormal basis -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		For finite-dimensional spaces the condition of a dense span is the same as 'span', as used in (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra.
		An orthonormal basis of a vector space V makes no sense unless V is given an inner product; (Click link for more info and facts about Banach space) Banach spaces do not generally have orthonormal bases.
		This is fundamental to the study of (The sum of a series of trigonometric expressions; used in the analysis of periodic functions) Fourier series.
	www.absoluteastronomy.com /encyclopedia/o/or/orthonormal_basis.htm (603 words)

	3d (Site not responding. Last check: 2007-11-07)
		For sets which are of a complicated structure, especially fractals, the Hausdorff dimension is useful.
		The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values.
		The Krull dimension of a commutative ring is defined to be the maximal length of a strictly increasing chain of prime ideals in the ring.
	www.yourencyclopedia.net /3d.html (629 words)

	Hamel (Site not responding. Last check: 2007-11-07)
		Hamel was appointed Professor of Mechanics at the German Technical University of Brünn on 3 October 1905.
		Hamel was appointed to the chair of mechanics at the Rheinisch-Westfälische Hochschule in Aachen on 1 October 1912, then in 1919 he moved to the Technical University of Charlottenburg in Berlin where he was appointed as professor of mathematics and mechanics.
		Hamel was clearly associated with the views of National Socialism and in 1933 spoke of a spiritual bond between mathematics and the "Third Reich".
	www.gap-system.org /~history/Mathematicians/Hamel.html (926 words)

	Station Information - Hamel dimension
		It is sometimes called Hamel dimension when it is necessary to distinguish it from other types of dimension.
		Every basis of a vector space has equal cardinality and so the Hamel dimension of a vector space is uniquely defined.
		However, the Hamel dimension depends on the base field, so while R has dimension 1 when considered as a vector space over itself, it has dimension c (the cardinality of the continuum) when considered as a vector space over Q (the rationals).
	www.stationinformation.com /encyclopedia/h/ha/hamel_dimension.html (184 words)

	Dimension article - Dimension Latin degrees freedom measurements shape Physical dimensions - What-Means.com (Site not responding. Last check: 2007-11-07)
		For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis.
		This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.
		The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.
	www.what-means.com /encyclopedia/Three-dimensional (596 words)

	ORTHONORMAL BASES (Site not responding. Last check: 2007-11-07)
		In mathematics, an orthonormal basis of an inner product space (i.e., a vector space with an inner product), or in particular of a Hilbert space, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is of length 1.
		Note that an orthonormal basis is not generally a "basis", i.e., it is not generally possible to write every member of the space as a linear combination of finitely many members of an orthonormal basis.
		Hamel bases are of little if any interest in inner product spaces, but realizing that a Hamel basis is something that an orthonormal basis is not, may shed some light on what an orthonormal basis is.
	www.websters-online-dictionary.org /definition/ORTHONORMAL+BASES (398 words)

	Basis (linear algebra) (Site not responding. Last check: 2007-11-07)
		In an infinte-dimensional Hilbert space, a set of vectors orthogonal to each other can never span the whole space via finite linear combinations, but what is called an orthonormal basis is a set of mutually orthogonal unit vectors that "span" the space via sometimes-infinite linear combinations.
		To better distinguish these notions, vector space bases are also called Hamel bases and the vector space dimension is also known as Hamel dimension.
		An "orthonormal basis" of an infinite-dimensional Hilbert space is not a Hamel basis
	www.sciencedaily.com /encyclopedia/basis__linear_algebra_ (838 words)

	Hamel basis (Site not responding. Last check: 2007-11-07)
		The phrase Hamel basis is sometimes used to denote a basis as defined above, in which the fact thatall linear combinations are finite is crucial.
		A set B is a Hamel basis of a vector space V if everymember of V is a linear combination of just finitely many members of B.
		Hamel basesof spaces of this kind are of little if any interest; orthonormal bases of these spaces are important to Fourier analysis.
	www.therfcc.org /hamel-basis-195558.html (754 words)

	Executive Thought Leadership Quarterly: 2000: October: Part 1: Q&A with Business Visionary Gary Hamel: Make the Future ... (Site not responding. Last check: 2007-11-07)
		Hamel: In a world where employees give more of their lives to their job than they give to their community, their family, or their faith, companies need to create places where people can bring all of their humanity to work, not just one piece of it.
		Hamel, founder and chair of Strategos and visiting professor of strategic and international management at the London Business School, urges companies and individuals alike to adopt an "innovation agenda" that welcomes new ways of thinking and acting on a regular basis.
		Hamel says the fundamental challenge companies face is reinventing themselves and their industries not just in times of crisis but continually.
	newsroom.cisco.com /dlls/tln/newsletter/2000/october/part1.html (1376 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		In that context, the "usual" basis is called a Hamel basis: it is a collection of vectors {v_i}, i in some index set, such that every vector in the space can be written uniquely as a linear combination of some finite set of v_i's.
		For a Schauder basis, on the other hand, you allow an infinite "linear combination", where the infinite sum is understood as an infinite series, converging in the norm topology of the Banach space.
		Hamel bases always exist, but can't be constructed without the Axiom of Choice.
	www.math.niu.edu /~rusin/papers/known-math/96/schauder (766 words)

	Beaumont Hamel (Site not responding. Last check: 2007-11-07)
		Hamel's Amusement Park 1: half, and in 1999 it closed permanently.
		Hamel, Illinois 1: '''Hamel ''' is a village located in Madison County, Illi 4: Hamel is located at 38anddeg;53'15" North, 89anddeg;50'38"
		Gary Hamel 1: '''Gary Hamel ''' is the CEO of Strategos, the managem 10: category:Business theoristsHamel, Gary
	www.daikaiju.com /edge/3883-beaumont%20hamel.html (263 words)

	PlanetMath: Banach spaces of infinite dimension don't have a countable algebraic basis (Site not responding. Last check: 2007-11-07)
		A Banach space of infinite dimension does not have a countable algebraic basis.
		Cross-references: metric, induced, metric space, complete, norm, Hamel basis, operations, vector space, sequences, QED, Baire category theorem, first category, every finite dimensional proper subspace of a normed space is nowhere dense, finite, subspaces, union, mean, iff, linear combination, countable, dimension, infinite, Banach space
		This is version 10 of Banach spaces of infinite dimension don't have a countable algebraic basis, born on 2005-01-31, modified 2005-09-18.
	planetmath.org /encyclopedia/ABanachSpaceOfInfiniteDimensionDoesntHaveACountableAlgebraicBasis.html (448 words)

	Home (Site not responding. Last check: 2007-11-07)
		Hamel Enterprises is a small software development firm with a twist.
		Hamel Enterprises is currently focusing on providing software services to the US Intelligence Community and Military Intelligence organizations by partnering/teaming and subcontracting with major US Government Prime Contractors.
		Hamel Enterprises is a member of the Virginia Fairfax Chamber of Commerce and RMP Consulting Partners.
	www.hamel-enterprises.com (287 words)

	Hilbert space - Wikipedia, the free encyclopedia
		One goal of Fourier analysis is to write a given function as a (possibly infinite) sum of multiples of given base functions.
		This problem can be studied abstractly in Hilbert spaces: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these base elements.
		Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis.
	en.wikipedia.org /wiki/Hilbert_space (1813 words)

	Orthonormal basis
		In mathematics, an orthonormal basis of an inner product space V(i.e., a vector space with an inner product), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is of length 1.
		If B is an orthonormal basis of H, then every element x of H may be written as
		Using Zorn's lemma, one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality.
	www.brainyencyclopedia.com /encyclopedia/o/or/orthonormal_basis.html (565 words)

	[No title]
		Hamel received cash and stock kickbacks totalling approximately $234,000.-[3]- In April 1993, transactions in Fairmont accounted for 60% of Hamel's overall trades.
		On March 14, 1995, a permanent injunction was entered, enjoining Hamel from future violations of 5(a), 5(c) and 17(a) of the Securities Act, 10(b) of the Exchange Act and Rule 10b-5 thereunder.
		Hamel told Laurienti that the accounts opened on April 23, 1993 had been referred to him by a registered representative at another firm who had been prohibited from selling Fairmont.
	www.sec.gov /litigation/admin/3436338.txt (2000 words)

	Video Game History - The Complete History of Video Games: Dimension (Site not responding. Last check: 2007-11-07)
		Hamel Dimension For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis.
		Manifolds A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension.
		Krull dimension of commutative rings The Krull dimension of a commutative ring is defined to be the maximal length of a strictly increasing chain of prime ideals in the ring.
	www.videogamehistory.info /Video_games/3d.shtml (487 words)

	Basis (linear algebra) - FreeEncyclopedia (Site not responding. Last check: 2007-11-07)
		A basis of a vector space is sometimes called a Hamel basis in order to distinguish it from the concept of an orthonormal basis of a Hilbert space and some other kinds of bases that occur in Banach spaces.
		An orthonormal basis of a Hilbert space H is an orthonormal set of members of H such that any member of the H can be written as a linear combination of a possibly infinite set of members of the orthonormal basis.
		Every basis of a vector space has the same cardinality, called the dimension of the vector space.
	openproxy.ath.cx /ba/Basis_(linear_algebra).html (363 words)

	PlanetMath: axiom of choice (Site not responding. Last check: 2007-11-07)
		Thus objects that are proved to exist using the axiom of choice cannot generally be described by any kind of systematic rule, for if they could it would not be necessary to their construction.
		Strange objects that can be constructed using the axiom of choice include non-measurable sets (leading to the Hausdorff and Banach-Tarski paradoxa), and Hamel bases for any vector space.
		Since in fact the existence of a basis for every vector space is equivalent to the axiom of choice, it is almost guaranteed that no such set
	planetmath.org /encyclopedia/AxiomOfChoice.html (639 words)

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